Note that two length functions on a locally compact group are equivalent if and only if the corresponding pre-metric spaces are quasi-isometric. In fact, this kind of equivalence is rather weak (but there are weaker ones). In particular, it cannot see the difference between the length functions induced by a left-invariant Riemannian metric and a Carnot-Carathéodory metric (which is sub-Riemannian) on a simply connected nilpotent Lie group; see [1]}.
